from syllogism to common sense: a tour through the logical landscape ... · a tour through the...

From syllogism to common sense:a tour through the logical landscape

Categorical syllogisms – Part 2

Mehul Bhatt Oliver Kutz Thomas Schneider

17 November 2011

Thomas Schneider Categorical syllogisms

What happened so far?

Part I


Thomas Schneider Categorical syllogisms What happened so far?


Categorical propositions (CPs)

. . . state relations between classes:

Example General form NameAll policitians are liars. All S is P. A Universal affirmationNo policitians are liars. No S is P. E Universal negationSome policitians are liars. Some S is P. I Particular affirmationSome policitians are not liars. Some S is not P. O Particular negation

. . . can be interpreted via Venn diagrammes:

All S is P. No S is P. Some S is P. Some S is not P.S P

x

S P

x

S P

x

S P

x



Categorical propositions (CPs)

. . . have a quantity, quality and distribution:

Proposition Quantity Quality DistributesAll S is P. A Universal Affirmative SNo S is P. E Universal Negative S,PSome S is P. I Particular Affirmative —Some S is not P. O Particular Negative P

Distribution: “Does the proposition make a statement about allmembers of S or P?”



Immediate inferences

. . . via square of opposition:

ContradictoriesCon

tradictor

ies

A E

I O

Subalternation Subalternation

Contraries

Subcontraries

Further inferences:Conversion (S 7→ P, P 7→ S; not always successful)

Obversion (Change quality, P 7→ non-P)

Contraposition (S 7→ non-P, P 7→ non-S; not always succ.)



Aristotelian versus Boolean interpretationAristotelian interpretation assumes existential import:S is nonempty

A All S is P. E No S is P.S P

x

S P

x

Boolean interpretation rejects existential import:in A and E, S may be empty

A All S is P. E No S is P.S P S P


Standard-form CSs Venn-diagramme technique Rules and fallacies The valid CSs Summary and outlook

Part II

Categorical syllogisms (CSs)

Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 36


In this part . . .

2 Standard-form CSs

3 Venn-diagramme technique for testing CSs

4 Rules and fallacies

5 The valid CSs

6 Summary and outlook



And now . . .

2 Standard-form CSs



5 The valid CSs




Basic notionsAim: more extended reasoning with CPs

Syllogism: deductive argument with 2 premises and 1 conclusion

Categorical syllogism:syllogism based on CPsdeductive argument of 3 CPsall 3 CPs together contain 3 termsevery term occurs in 2 propositions

Syllogisms are common, clear and easily testable. They areone of the most beautiful and also one of the mostimportant made by the human mind.

(GOTTFRIED WILHELM LEIBNIZ, 1646–1716, German philosopherand mathematician, Hannover)



Standard-form CSs

1 Premises and conclusion are standard CPs (A, E, I, O)2 CPs are arranged in standard order:

. . . S1 is . . . P1

. . . S2 is . . . P2∴ . . . S is . . . P

P: major term, S: minor term

Remember: 3 terms altogether, each in 2 propositions!; S1, S2,P1, P2 consist of P, S and a third term: the middle term

Major premise contains P,MMinor premise contains S,M



Examples

Major term, minor term, middle term

All great scientists are college graduates.Some professional athletes are college graduates.Therefore some professional athletes are great scientists.

All artists are egotists.Some artists are paupers.Therefore some paupers are egotists.



Mood of a CS

Mood of a CS is the pattern of types of its three CPs,in the order major premise – minor premise – conclusionsA All artists are egotists.I Some artists are paupers.I Therefore some paupers are egotists.

Mood AII

; 43 = 64 moods

Thomas Schneider Categorical syllogisms Categorical syllogisms (CSs) 6 · 9


Figure of a CS

Figure of a CS: combination of order of S,M,P in the premises:

No P is M P–MSome S is not M has figure S–M

∴ All S is P ∴ S–P

; 4 figures:(1) M–P

S–M∴ S–P

(2) P–MS–M

∴ S–P

(3) M–PM–S

∴ S–P

(4) P–MM–S

∴ S–P



Formal nature of the syllogistic argument

There are only 4 · 64 = 256 possible forms of CSs.

Their validity can be exhaustively analysed and established.

Only a few will turn out to be valid.

Infinitely many (in-)valid syllogistic arguments can be obtainedby replacing S,M,P in a(n in-)valid CS with “real-world” classdescriptions.



And now . . .

2 Standard-form CSs



5 The valid CSs




Testing a form of CS for validity

. . . is very simple!

1 Draw three overlapping cycles for S,P,M:

S P

M

2 Mark the premises according to their types as earlier.

E.g.: AAA-1 All M is P.All S is M.

∴ All S is P.

S P

M

3 Try to read off the conclusion without further marking.Syllogism type is valid iff reading off was successful.



Example

What form does this syllogism have? Is it valid?

All dogs are mammals.All cats are mammals.Therefore all cats are dogs.



Two cautions

(1) Mark universal before particular premise.

All artists are egotists.Some artists are paupers.Therefore some paupers are egotists.

Paupers Egotists

Artists

x

(2) If a particular premise speaks about two nonempty regions,put the x on the boundary of these regions.

All great scientists are college graduates.Some professional athletes are college graduates.Therefore some professional athletes are great scientists.

PA GS

CG

x



Examples

AEE-1 All M is P.No S is M.

∴ No S is P.

S P

MInvalid: diagramme does not exclude S from P.

EIO-4 No P is M.Some M is S.

∴ Some S is not P.

S P

M

x

Valid: diagramme gives a particular instance of S \ P.



And now . . .

2 Standard-form CSs



5 The valid CSs




An alternative characterisation of validity of CSs. . . via rules that focus on the form of the syllogism

Rule 1: Avoid four terms.With > 4 terms, it’s no syllogism at allBeware of equivocations!(two occurrences of the same word with different meanings)

And the Lord spake, saying, “First shalt thou take out theHoly Pin. Then, shalt thou count to three. No more. No less.Three shalt be the number thou shalt count, and the numberof the counting shall be three. Four shalt thou not count,neither count thou two, excepting that thou then proceed tothree. Five is right out. Once at the number three, being thethird number to be reached, then, lobbest thou thy Holy HandGrenade of Antioch towards thy foe, who, being naughty inMy sight, shall snuff it.”(from “Monty Python and the Holy Grail”, 1975)



Distribute your middle term

Rule 2: Distribute the middle term in at least one premise.(One proposition must refer to all members of M.)Example: All Russians were revolutionists.

All anarchists were revolutionists.Therefore all anarchists were Russians.

Fallacy: middle term “revolutionists” doesn’t link S,PRussians are included in a part of revolutionistsAnarchists are included in a part of revolutionists,possibly a different part!

Fallacy of the undistributed middle



Watch your distribution

Rule 3: Any term distributed in the conclusion must be distributedin the premises.Intuition: if premises speak about some members of a class,we cannot conclude anything about all members of that class.Example: All dogs are mammals.

No cats are dogs.Therefore no cats are mammals.

Fallacy: “mammals” is distributed in the conclusion,but not in the major premise.

Fallacy of illicit process (here: illict process of the major term)



Two negative premises are bad

Rule 4: Avoid two negative premises.2 negative premises; 2× class exclusion between S,M and between P,M

No power to enforce any relation between S,P

Try all nine possibilities in a Venn diagramme!

Example: No artists are accountants.Some poets are not accountants.Therefore some poets are not artists.

Fallacy of exclusive premises



Don’t turn neg into pos

Rule 5: If > 1 premise is negative, the conclusion must be neg.

Affirmative conclusion =̂ one of S,P is (wholly or partly)contained in the other.Can only be inferred if premises assert existence of Mwhich contains one of S,P and is contained in the otherClass inclusion only via affirmative propositions

Example: No poets are accountants.Some artists are poets.Therefore some artists are accountants.

Fallacy of drawing an affirmative conclusion from a neg. premise



Don’t be so Aristotelian

Rule 6: From two universal premises, no particular conclusion maybe drawn.Example: All household pets are domestic animals.

No unicorns are domestic animals.Therefore some unicorns are not household pets.

Existential fallacy (not a fallacy in the Aristotelian interpretation)



And now . . .

2 Standard-form CSs



5 The valid CSs




The 15 valid forms of syllogisms

AAA-1 Barbara AII-3 DatisiEAE-1 Celarent IAI-3 DisamisAII-1 Darii EIO-3 FerisonEIO-1 Ferio OAO-3 BokardoAEE-2 Camestres AEE-4 CamenesEAE-2 Cesare IAI-4 DimarisAOO-2 Baroko EIO-4 FresisonEIO-2 Festino



And now . . .

2 Standard-form CSs



5 The valid CSs




Summary

Categorical syllogisms . . .are deductive arguments consisting of 3 CPsrequire a certain amount of interactionbetween the terms in their CPscome in 4 figures and 64 moodscan be tested for validity using Venn diagrammesor rules/fallacies

There are 15 valid forms of syllogisms in Boolean interpretation,24 in Aristotelian interpretation

It’s almost play time:http://www.theotherscience.com/syllogism-machine

Try with examples from Pages 41 47 53 55 56


http://www.theotherscience.com/syllogism-machine


Literature and outlook

Contents is taken from Chapters 5, 6 of

I. Copi, C. Cohen, K. McMahon:Introduction to Logic, 14th ed., Prentice Hall, 2011.

SUUB Magazin 02 E 2115Link to available copies

Chapter 7:transform arguments of everyday speech into syllogistic form,possible difficulties

Thank you.


http://www.suub.uni-bremen.de/

http://suche.suub.uni-bremen.de/cgi-bin/CiXbase/brewis/CiXbase_search?act=search&INDEXINFO=awCN&LAN=DE&ORDER=ID&IHITS=15&FHITS=15&PRECISION=220&RELEVANCE=55&n_dtyp=1L&n_rtyp=ceEdX&index=L&XML_STYLE=%2Fstyles%2Fcns-DE.xml&dtyp=a&mtyp=&section=&term=copi+introduction+to+logic&CID=1749606&x=0&y=0

from syllogism to common sense: a tour through the logical landscape ... · a tour through the...

Documents